6 A quantum game

Let’s play a game. We form two teams of three, the “players” (Andy, Bob, and Charles) and the “interrogators”.[1] The rules of this game are as follows:

Either all players are asked for the value of X, or one player is asked for the value of X while the two other players are asked for the value of Y.

The possible values of both X and Y are +1 and −1.

If all players are asked for the value of X, they win if (and only if) the product of their answers equals −1. Otherwise they win if (and only if) the product of their answers equals +1.

Once the questions are asked, the players are no longer allowed to communicate with each other. Prior to that, they may work out a strategy. Is there a fail-safe strategy? Can they make sure that they will win? Ponder this before you proceed.

The obvious strategy is to use pre-agreed answers.

Let’s call them XA, XB, XC, and YA, YB, YC.

Now try this: Assign values (+1 or −1) to the following variables in such a way that the product of the three X values equals −1 while the product of the Y values in two of the three columns equals the X value in the remaining column — or else explain why this can’t be done.
XA XB XC
YA YB YC